Thank you for the information and your question. Although your employer is mistaken, or not being quite honest, about the requirements of OSHA (not allowing an employee to bring their own chair), that really doesn't mean the employer can't prohibit employees from bringing their own furniture, including chairs. In other words, OSHA actually doesn't have any specific rules about chairs or desks, but instead only has the general rule that an employer must keep the workplace free from all hazards, including ergonomic ones. However, an employer always has the right to set the terms and conditions of employment (absent a bargaining agreement that states differently), including all workplace rules. So, if your employer says no to the chair, they have the legal right to do so.
All that said, you have the right, if your doctor says you need a specific type of chair, to submit that information to your employer, as they relayed to you, so that they can purchase a chair that will fit your physical requirements.
Please let me know if you need any clarification. If none is needed, then if you could take a moment to leave a positive rating in the box above, I will receive credit for assisting you today. Thank you